Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Average Error: 0.0 → 1.3

Time: 4.2s

Precision: binary64

Cost: 6660

Math

\[e^{\left(x + y \cdot \log y\right) - z} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 3.35 \cdot 10^{-15}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- (+ x (* y (log y))) z)))

↓

(FPCore (x y z) :precision binary64 (if (<= z 3.35e-15) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.35e-15) {
		tmp = exp(x);
	} else {
		tmp = exp(-z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.35d-15) then
        tmp = exp(x)
    else
        tmp = exp(-z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.35e-15) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

↓

def code(x, y, z):
	tmp = 0
	if z <= 3.35e-15:
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.35e-15)
		tmp = exp(x);
	else
		tmp = exp(Float64(-z));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.35e-15)
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[z, 3.35e-15], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Target

Original	0.0
Target	0.0
Herbie	1.3

\[e^{\left(x - z\right) + \log y \cdot y} \]

Derivation

1. Split input into 2 regimes

2. if z < 3.35e-15

   1. Initial program 0.0

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Simplified 11.0

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z - x}}} \]

   3. Taylor expanded in z around 0 11.5

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{-x}}} \]

   4. Taylor expanded in y around 0 1.6

      \[\leadsto \color{blue}{\frac{1}{e^{-x}}} \]

   5. Simplified 1.6

      \[\leadsto \color{blue}{e^{x}} \]

   if 3.35e-15 < z

   1. Initial program 0.0

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 0.9

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]

   3. Simplified 0.9

      \[\leadsto e^{\color{blue}{-z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.35 \cdot 10^{-15}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.9
Cost	6464

\[e^{x} \]

Alternative 2
Error	44.5
Cost	192

\[x + 1 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (exp (+ (- x z) (* (log y) y)))

  (exp (- (+ x (* y (log y))) z)))